Pseudo-Riemannian manifold

Definition

Let $M$ be a finite-dimensional smooth manifold. A pseudo-Riemannian metric on $M$ is a smooth symmetric section

g\in\Gamma^\infty(T^*M\otimes T^*M)

such that $g_p$ is nondegenerate on $T_pM$ for every $p\in M$ . A pseudo-Riemannian manifold is a pair $(M,g)$ . On each connected component, the numbers of negative and positive squares in a diagonalization of $g_p$ are constant; this ordered pair is the signature of $g$ .

Local form and convention

In local coordinates,

g=g_{ij}\,dx^i\otimes dx^j,

where $(g_{ij})$ is an invertible symmetric matrix at every point. This knowl writes the signature as $(r,s)$ , with $r$ negative and $s$ positive directions. Thus a metric of signature $(1,n-1)$ is locally represented by $\operatorname{diag}(-1,1,\ldots,1)$ .

Nondegeneracy, rather than positive definiteness, is the essential condition. It makes the bundle map $g^\flat:TM\to T^*M$ an isomorphism and gives an inverse metric $g^{-1}$ on $T^*M$ .

Canonical constructions

Exactly as in Riemannian geometry, $g$ determines a unique torsion-free metric-compatible Levi–Civita connection. It also determines curvature and a volume density. Unlike a Riemannian metric, an indefinite metric does not define a norm or metric-space distance: nonzero vectors can be null, and the quadratic form can take either sign.

Important special cases

Signature $(0,n)$ gives a Riemannian metric in this convention. Signature $(1,n-1)$ gives a Lorentzian manifold. Reversing the overall sign exchanges $(r,s)$ and $(s,r)$ without changing the null cone, so both sign conventions occur in the literature.

References

Barrett O'Neill, Semi-Riemannian Geometry With Applications to Relativity, Academic Press, 1983. Publisher record. Relevant: Chapters 1–3.
John K. Beem, Paul E. Ehrlich, and Kevin L. Easley, Global Lorentzian Geometry, 2nd ed., Marcel Dekker, 1996. Publisher record. Relevant: Chapter 1.